Simple inexpensive vertex and edge invariants distinguishing dataset strongly regular graphs

TL;DR

The paper tackles the graph isomorphism problem for strongly regular graphs by introducing inexpensive polynomial-time invariants that operate directly on the original graphs. It develops vertex invariants from neighborhood-restricted powers of the adjacency matrix $A$ via $\tilde{A}$ and edge invariants from edge-pair matrices $\bar{A}$, using traces (and in some cases sorted diagonals) of powers to obtain rotation-invariant discriminants. On Edward Spence's SRG dataset of $v$-vertex graphs, the vertex invariants distinguish all but 4 graph pairs, with the remaining pairs separated by the edge invariants, demonstrating strong empirical performance in a notoriously hard GI class. The results hint at a potential polynomial-time approach to GI for SRGs and motivate further work toward formal completeness proofs and broader applicability to other graph families.

Abstract

While standard Weisfeiler-Leman vertex labels are not able to distinguish even vertices of regular graphs, there is proposed and tested family of inexpensive polynomial time vertex and edge invariants, distinguishing much more difficult SRGs (strongly regular graphs), also often their vertices. Among 43717 SRGs from dataset by Edward Spence, proposed vertex invariants alone were able to distinguish all but 4 pairs of graphs, which were easily distinguished by further application of proposed edge invariants. Specifically, proposed vertex invariants are traces or sorted diagonals of $(A|_{N_a})^p$ adjacency matrix $A$ restricted to $N_a$ neighborhood of vertex $a$, already for $p=3$ distinguishing all SRGs from 6 out of 13 sets in this dataset, 8 if adding $p=4$. Proposed edge invariants are analogously traces or diagonals of powers of $\bar{A}_{ab,cd}=A_{ab} A_{ac} A_{bd}$, nonzero for $(a,b)$ being edges. As SRGs are considered the most difficult cases for graph isomorphism problem, such algebraic-combinatorial invariants bring hope that this problem is polynomial time.

Figures (4)

• Figure 1: Example of proposed inexpensive vertex invariants: just sorted diagonals of $(A|_{N_a})^4$, combinatorially numbers of length 4 closed paths inside neighborhood of vertex $a$. There is shown used Mathematica code (top) and adjacency matrices on the left of sorted lexicographically visualized with colors calculated vertex invariants: $25\times 12$ matrices being graph invariants for 15 SRGs of 25-12-5-6 parameters, allowing to distinguish all of them as non-isomorphic. These vertex invariants usually allow to split vertices into subsets of constant invariants, marked by blue lines - allowing to restrict potential automorphisms to those applying permutations inside such subsets.
• Figure 2: The only 4 pairs of graphs in the dataset for which the discussed vertex invariants were insufficient to distinguish - in contrast to the presented further edge invariant. For "inblock" two of them (left) the vertex invariants were not able to split vertices into subsets (happened to 39 of 43717 SRGs shown in Fig. \ref{['unique']}), and such invariants were identical for both graphs. For "outblock" two of them (right), beside sorted $\textrm{diag}((A|_{N_a})^p)$ vertex invariants for the entire graph, if needed there were also used such invariants for subgraph with removed first set of distinguished vertices - leaving the shown two pairs, having subtle differences of connections between its subgraphs (marked yellow). Grayness shows split of edges into subsets of constant values of $\textrm{diag}(\bar{A}^5)$.
• Figure 3: Adjacency matrices of all 39 SRGs in the dataset (of 43717 SRGs) for which the discussed vertex invariants were not able to distinguish vertices (e.g. to restrict automorphisms, they can still differ between graphs allowing to distinguish graphs). Marked two pairs, shown also in Fig. \ref{['problem']}, are the only two requiring further edge invariants to be distinguish - they could also split edges into conserved subsets of the same invariants, marked with grayness.
• Figure 4: Analogously as in Fig. \ref{['d25']}, but for $p=3$ power and 41 SRGs of 29-14-6-7 parameters - allowing to distinguish them with sorted $\textrm{diag}((A|_{N_a})^3)$ vertex invariants, also distinguish some vertices: splitting them into blocks with shown blue lines.